| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s + 54·5-s − 36·6-s − 64·8-s + 81·9-s − 216·10-s + 216·11-s + 144·12-s − 998·13-s + 486·15-s + 256·16-s − 1.30e3·17-s − 324·18-s − 884·19-s + 864·20-s − 864·22-s − 2.26e3·23-s − 576·24-s − 209·25-s + 3.99e3·26-s + 729·27-s − 1.48e3·29-s − 1.94e3·30-s − 8.36e3·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.965·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.683·10-s + 0.538·11-s + 0.288·12-s − 1.63·13-s + 0.557·15-s + 1/4·16-s − 1.09·17-s − 0.235·18-s − 0.561·19-s + 0.482·20-s − 0.380·22-s − 0.893·23-s − 0.204·24-s − 0.0668·25-s + 1.15·26-s + 0.192·27-s − 0.327·29-s − 0.394·30-s − 1.56·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 54 T + p^{5} T^{2} \) |
| 11 | \( 1 - 216 T + p^{5} T^{2} \) |
| 13 | \( 1 + 998 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1302 T + p^{5} T^{2} \) |
| 19 | \( 1 + 884 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2268 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1482 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8360 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4714 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9786 T + p^{5} T^{2} \) |
| 43 | \( 1 - 452 p T + p^{5} T^{2} \) |
| 47 | \( 1 + 22200 T + p^{5} T^{2} \) |
| 53 | \( 1 - 26790 T + p^{5} T^{2} \) |
| 59 | \( 1 + 28092 T + p^{5} T^{2} \) |
| 61 | \( 1 - 38866 T + p^{5} T^{2} \) |
| 67 | \( 1 - 23948 T + p^{5} T^{2} \) |
| 71 | \( 1 + 20628 T + p^{5} T^{2} \) |
| 73 | \( 1 + 290 T + p^{5} T^{2} \) |
| 79 | \( 1 + 99544 T + p^{5} T^{2} \) |
| 83 | \( 1 + 19308 T + p^{5} T^{2} \) |
| 89 | \( 1 + 36390 T + p^{5} T^{2} \) |
| 97 | \( 1 - 79078 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.13138878422012828275125896465, −9.479238417332253732506700943464, −8.805146359968703072457581298369, −7.59686932792248376289971633177, −6.74287202410047817896836804514, −5.59745284526236769814277142920, −4.15819202142693109125843893692, −2.48258085168926064275249611604, −1.81189726127868307563972767213, 0,
1.81189726127868307563972767213, 2.48258085168926064275249611604, 4.15819202142693109125843893692, 5.59745284526236769814277142920, 6.74287202410047817896836804514, 7.59686932792248376289971633177, 8.805146359968703072457581298369, 9.479238417332253732506700943464, 10.13138878422012828275125896465
